The idea of the existence of ideal free particles (FP) is rooted in the philosophy of Descartes, who had prescribed eliminating “the influences from far away”to remove the plague of medieval sciences [1]}. Modern science adopted the concepts of the existence of free particles in a similar fashion [2]}. The notion of FP is widely accepted to be fruitful for the approximate behavior of a physical system. However, the quantum entanglement phenomenon seems to defy the strict meaning of FP [3]}. Therefore, it seems reasonable to claim that the problem of FP deserves attention in its own right. In this article, we have revisited the aspects of a time-dependent effective mass, which is moving as an FP.
The time-dependent effective mass (TDEM) appears to be relevant for a diverse domain of Physics, namely in the energy density functional approach to many-body problems [4]}, [5]}, in the study of electronic properties of condensed matter systems [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, the Schrödinger equation in curved space under the shed of deformed algebras [14]}, nonlinear optical properties in quantum well [15]}, [16]}, and even the cosmological models to quantum information theory [17]}, [18]}, [19]}, [20]}, [21]}, [22]} and many more [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, [30]}. For instance, the “asymmetric shape of crackling noise pulses emitted by a diverse range of noisy systems”is modeled with the help of TDEM in [31]}, [32]}. For a TDEM, the system becomes non-conservative [33]}, [34]}. In particular, if a quantum system interacts with time-varying environments such as temperature, pressure, stress, and energy, the effective masses will be modified by some TDEM [33]}, [34]}, [37]}, [38]}, [39]}, [40]}.
A first principle calculation for the viable Hamiltonian of a position and time-dependent effective mass was proposed in [41]}.
The non-perturbative solution of the time-dependent Schrödinger equation (TDSE) for non-conservative systems (in particular for TDEM) is a tricky one (if not impossible). However, if a class of phase-space invariant operators (PSIO) corresponding to the time-dependent Hamiltonian (\(\hat{H}(t)\) ) exists, then the system is exactly solvable. For example, the quadratic (both in position and momentum) Hamiltonians are solvable by some factorization method [42]}, [43]}, [44]}.
